Mean (arithmetic average)

The mean (or arithmetic average) is the single most familiar way to summarise a list.

Idea

To find the mean of a list, add up all the values and divide by the count: $\text{Mean} = \frac{\text{sum of values}}{\text{number of values}}.$

For example, mean of $4, 6, 8$ is $\tfrac{4+6+8}{3} = \tfrac{18}{3} = 6$.

Why the mean? If you replaced every value in the list with the mean, the total would be the same. So the mean is the value that "evens out" the data.

A small example. Three classmates spend $\rupee 30, \rupee 50, \rupee 70$ on snacks. Mean: $\rupee 50$. If each had spent exactly $\rupee 50$, the total would still be $\rupee 150$. So the mean is the fair-share value.

Outliers. A single extreme value can yank the mean drastically. Consider $1, 2, 3, 4, 100$. Mean $= \tfrac{110}{5} = 22$. But $22$ is bigger than four of the five values! The outlier $100$ has pulled the mean up. In such cases, the median (next subtopic) gives a better picture.

Mean is good for balanced data, scores on a test, averages over time.

Worked examples

Example 1. Find the mean of $5, 10, 15$.

$\tfrac{5+10+15}{3} = \tfrac{30}{3} = 10$.

Example 2. Find the mean of $12, 18, 14, 20, 16$.

$\tfrac{12+18+14+20+16}{5} = \tfrac{80}{5} = 16$.

Example 3. Five test scores: $7, 8, 9, 6, 10$. Mean?

$\tfrac{7+8+9+6+10}{5} = \tfrac{40}{5} = 8$.

Example 4. Mean of $4$ numbers is $20$. Find their total.

Total $=$ mean $\times$ count $= 20 \times 4 = 80$.

Try it yourself

1. Mean of $4, 8, 12, 16$.
2. Mean of $5, 7, 11, 13$.
3. Five students scored $80, 75, 90, 85, 70$. Mean?
4. Mean of $3$ consecutive integers starting from $10$.
5. The mean of $5$ numbers is $12$. Find the total.
6. Daily temperatures: $28, 30, 29, 31, 32$. Mean?
7. Mean of $1$ through $10$. (Use the formula $\tfrac{n+1}{2}$ for $1$ to $n$.)
8. Pocket money of $4$ siblings: $\rupee 50, 60, 70, 100$. Mean?

Activity

Find the mean height of your $5$ friends (in cm). Compare with your own height.